The ensemble of gene regulatory networks at mutation-selection balance

The evolution of diverse phenotypes both involves and is constrained by molecular interaction networks. When these networks influence patterns of expression, we refer to them as gene regulatory networks (GRNs). Here, we develop a quasi-species model of GRN evolution. With this model, we prove that–across a broad spectrum of viability and mutation functions–the dynamics converge to a stationary distribution over GRNs. Next, we show from first principles how the frequency of GRNs at equilibrium will be proportional to each GRN’s eigenvector centrality in the genotype network. Finally, we determine the structural characteristics of GRNs that are favored in response to a range of selective environments and mutational constraints. Our work connects GRN evolution to quasi-species models, and thus can provide a mechanistic explanation for the topology of GRNs experiencing various evolutionary forces.